maximal positive boundary value problems as limits of singular ...

transactions of the american mathematical

society

Volume 270, Number 2, April 1982

MAXIMALPOSITIVE BOUNDARYVALUEPROBLEMS AS LIMITS OF SINGULARPERTURBATIONPROBLEMS1 BY CLAUDE BARDOS AND JEFFREY RAUCH Abstract. We study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain £2 C R". In all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of Friedrichs. The perturbation is either a second order elliptic term or a term large on the complement of Q. The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in Í2 and outside Ü. The results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in £2. The boundary condition is determined in a simple way from the system and the singular terms.

1. Introduction. In this paper we study boundary value roblems for singularly perturbed systems of partial differential equations. The unperturbed system is of first order and positive symmetric in the sense of Friedrichs. We study three sorts of singular perturbations: (1) the addition of e times a positive second order elliptic

system in a domain fi, (2,3) the addition of XP or \P(d/dt)

with À » 1 and P

strictly positive in the exterior of ñ and zero in S2. In the first case we study the solutions He to the Dirichlet problem in S2.Bardos, Brezis and Brezis [1] have shown that in this case ue^u weakly in L2(fi) where u is uniquely determined as the solution of a maximal positive boundary value problem for the unperturbed operator. Here one has the phenomenon of loss of boundary conditions in the limit and the existence of boundary layers for uc near 3Í2. We prove several more refined estimates on ue, in particular a uniform bound in //l/2_,,(fi) for any rj > 0. These estimates yield a proof of the strong convergence of uc to u in all Hs(ü) for s < {-. This singular perturbation problem is well known and much studied, in contrast to the problems of the second sort mentioned above. In the latter problems we study the Cauchy problems

(L + perturbation) ux - 0 u(0,-)

on [0, T] X R",

= g(x)

onR",

Received by the editors July 27, 1978 and, in revised form, August 8, 1980. 1980 Mathematics Subject Classification. Primary 35B25, Secondary 35F15, 35L50. Key words and phrases. Singular perturbations, boundary layers, maximal positive boundary value problems. 'Research supported by the National Science Foundation under grant number NSF MCS 79-01857. ©1982 American

Mathematical

Society

0002-9947/81 /0000- 1049/J08.S0

377

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378

CLAUDE BARDOS AND JEFFREY RAUCH

describing wave propagation on all of R". The perturbation is very large on the complement of a region [0, T] X ß. The phenomenon we find is that as X -> oo,

ux -» 0

outside [0, F] X ß,

ux -> u

in[0, F] X ß,

where m is uniquely determined as the solution of a maximal positive boundary value problem on [0, F] X ß with homogeneous boundary conditions on [0, F] X 3ß determined by the operator L and the matrix P on [0, T] X 3ß. Note that in the first problems we passed from one boundary value problem ß to a limiting boundary value problem on ß while in the present context one passes from a Cauchy problem on [0, F] X R" to a mixed initial boundary value problem on [0, F] X ß. In problems of wave propagation one often introduces mixed problems on a domain [0, T] X ß where the conditions on [0, F] X 3ß describe the interaction of waves with the boundary. Invariably such models involve idealizations about the containing walls [0, T] X 3ß. In reality waves always penetrate beyond the walls. A more complete model would involve an equation on [0, F] X R" which suffers an abrupt change at [0, F] X 3ß reflecting the fact that the medium in ß is very different from that in the walls. What is needed then is a theorem asserting that in [0, F] X ß the solution is approximated by solutions of the mixed problem which one has proposed as a model. Our theorems provide rigorous results of this type. This point of view is presented in an unpublished set of lecture notes by Friedrichs [4] where he predicts, by a formal asymptotic analysis, the results we prove. Notice that for all three singular perturbation problems considered above the limiting problem is a maximal positive boundary value problem in the sense of Friedrichs [3]. This has two important consequences. First, the limiting boundary value problems are well posed. This is a crucial fact, since in reality one never observes the limit X -» oo or e -» 0, only X very large or e very small. Then the quantities inside the region do not satisfy the homogeneous boundary condition Mu = 0 but do have Mu small. Since the boundary value problem is well set, it follows that u differs from the limiting solutions by an amount which can be estimated in terms of Mu. Thus if one observes that there is very little disturbance outside ß this is sufficient to justify the use of the limiting equations. From a practical point of view, this gives a criterion for knowing when to use the limiting equations. Second, a special place is assumed by the maximal positive boundary conditions among the many well-set boundary value problems. Since "physical" boundary conditions arise as limits when some parameter is large, we expect the basic boundary value problems of mathematical physics to be maximal positive. It should be remarked that, corresponding to the fact that our perturbations are strictly positive, the boundary conditions we find are not only positive but strictly positive. If the boundary matrix Av is nonsingular this means (Avu, u)> c\u\2 for all vectors u in the appropriate boundary space. In particular one does not find conservative boundary conditions, which satisfy (Avu, u) = 0, with these methods. There are a few well-understood problems which lead to conservative conditions: Dirichlet + Neumann problems for utt — A« and some problems in elasticity [4]. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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MAXIMALPOSITIVE BOUNDARY VALUE PROBLEMS

For these problems the positive definiteness of the perturbation is relaxed to positive semidefiniteness, and at present we do not have a good general theory encompassing these examples. On the other hand the inverse problem has a successful resolution in the category of strictly positive boundary conditions. That is, given a strictly positive boundary condition, there exist singular perturbations of all three kinds which yield this boundary condition in the limit. The interested reader is referred to [17] for the proof. Some of the problems we discuss have discontinuous coefficients, large in one region and not large in an adjacent region. One can argue that this too is an idealization and that one should replace such discontinuities by narrow regions where the coefficients change rapidly. Our methods can be adapted to treat such problems provided that in the appropriate limit the width of the narrow regions shrinks to zero rapidly. The details of this modification will not be described. An important ingredient in the proof of all of the theorems are estimates, uniform in the perturbation parameter, of the derivatives of solutions in directions parallel to the boundaries. This is consistent with the emergence of a boundary layer where quantities change rapidly as one moves away from the boundary and slowly as one moves parallel to the boundary (see [14, 15] for a discussion of boundary layers for problems analogous to ours). The paper is organized as follows. In §2 we give the notations and we state the theorems. Theorem 1 is concerned with the limit of vanishing viscosity. Theorems 2-4 deal with the results predicted by Friedrichs. In §3 we prove weak convergence for Theorems 2-4. §4 is devoted to the tangential regularity theorems whose proofs have a common thread. In all cases the idea is to take the given equation, differentiate tangentially and then apply the energy method. The crux is that by an appropriate change of independent and dependent variables one can cast the given differential operator into a special form so that the commutator with tangential differentiation is suitably small. Given these results the proof of strong convergence is not difficult, with the exception of the problem of vanishing viscosity. Here the proof of the //1/2 ^ estimate requires new ideas and is especially troublesome when 3ß is characteristic for the unperturbed system. The proofs of strong convergence and the Hx/2~n estimates are located in §5. Acknowledgment. In our research we have benefitted from the careful reading and thoughtful criticisms of J. Ralston. We offer him our hearty thanks.

2. Notations and statements of the results. Let L be the symmetric system of partial differential operators

(2.1)

L = ÍAJ(x)^i

oxj

+ B(x).

That is, Aj E C3(R") and B E C2(R") are A:X A:matrix-valued functions with Aj(x) hermitian symmetric for all x E R". We assume that DßB and DaAj forj = 1,...,«, | a | < 3, | ß | < 2, are uniformly bounded on R". We denote by

(2.2)

&,=-

inf minU:A

2 1ER"

G spectrum B(x) + B*(x) - Y ~^-\.

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OX,

380

CLAUDE BARDOS AND JEFFREY RAUCH

ß will denote an open set of R" which hes on one side of its smooth boundary 3ß. We do not assume that ß is bounded but we do assume that 3ß is bounded. A smooth boundary space N is a continuously differentiable map defined in a neighborhood of 3ß with value in Tp(Ck), the manifold of /^-dimensional linear

subspaces of C*. For any m X m hermitian matrix B we denote by B° the kernel of B, by B± the space spanned by the eigenvectors with strictly positive (negative) eigenvalues (Cm = B+ ®B° © B ). We denote by M(B) the space B+ ®B°. We assume that there is a C3 vector field v(x) defined in a neighborhood of 3ß with the following properties: (1) on 3ß, v(x) coincides with the unit normal to 3ß which points out of ß; (2) on a neighborhood of each connected component of 3ß there are matrix-valued functions U, a+ , a strictly positive and diagonal so that

(2.3)

del

Ar(x)=

, of class C3 with U unitary and ±a±

1a+(x)

2vj(x)Aj(x)

= U*(x)

0

0

a_(x)

0

0

0^

0 U(x).

0/

In particular, the dimensions of A°, A¿ , A~ and M(AV) are constant on each such

neighborhood. Concerning norms and inner products, \\\\x will denote the norm in the space X, | and ( , ) the norm and inner product in Ck and (, ) or (, )% the inner product in L2(%). We use the symbol L2(%) for both scalar and C*-valued L2 spaces. Thus

(S,SW= f(S,S)dx. Jen JelL A subspace N of C* is maximal positive for a matrix B if and only if

(2.4)

(Rtj,t])>0

(2.5)

V-nEN,

dimA = dimM(R).

The classical result is the following: Let N(x) be a smooth boundary space; assume that N(x) is for any x E 3ß a maximal positive subspace for Av, then (see [3, 8, 11]) for every / E L2(ß) and 8 > 80, there exists a unique u which is a solution of the following problem:

(2.6)

8u + Lu=f

inß,

u |3i2E N.

When (2.3) is satisfied, it is clear that M(A„) is a maximal positive smooth boundary space. Under the same hypothesis it is also true (see [1]) that if E(x) is a smooth positive hermitian matrix, E~x/2(x)(M(E-x/2(x)A

v(x)E-x/2(x)))

is a maximal positive subspace and therefore there exists for every / E L2(ß) a unique solution u E L2(ß) of the following problem: 8u + Lu=f,

F1/2w|3ß E m(e~x/2AvE-x/2).

Finally we recall some facts about traces (see [1]). If for each x E 3ß we write C* = Ker(A„) © (KerA,,)-1, then for any u E L2(ß) with Lu E L2(ß) the projection of u |3a on (KerA^)-1 is well defined as an element of //_,/2(3ß). Since License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

MAXIMALPOSITIVE BOUNDARY VALUE PROBLEMS

381

KerAy C N for any maximal positive subspace N, if Lu E F2(ß)/c the condition u |3Ö E N can be made precise in two equivalent ways. First using traces we may compute the projection of u |3S2onto N1- and insist that it vanish, and second, we

could demand that the relation

/ (t>, Lu) dx — I (L*v, m) dx hold for all v E C„\Jß) such that Ay(x)v(x) E N(x)1- for all x E 3ß. Here L* is the formal adjoint differential operator to L. Now we give the main statements. Theorem 1. Assume that 8 > 80 and that EtJ(x) is a family of n2 hermitian matrices with the following properties: (i) The matrices D"EU, I a | < 2, are bounded and continuous on a neighborhood o/ß. def

(ii) For every unit vector £ = (£,, £2,... ,£„) G R" the matrix F£ • £ = 2,7£,-,£,£, is uniformly strictly positive in ß. Then, for every f E L2(ß), the sequence ue of solutions to

(2-7)

-e2-^Eu-^ ¡j ■

+ 8ut + Lue=f,

j

«J8O= 0,

converges strongly in L2(fi), when e goes to zero to the solution of the following problem:

(2.8)

8u + Lu=f,

(Ewy/2u\düEM{(Evvy'/2AXEvv)-X/2).

Theorem 2. Let 8 > 80 and P a twice continuously differentiable hermitian symmetric-valued function on the complement, ßc, of ß and suppose all first order partial derivatives of P are bounded on ßc. We suppose that P is uniformly strictly positive on ßc. We denote also by P the function obtained by setting P(x) — 0 for x E ß; then for every f E L2(R"), the solution ux of the problem

(2.9)

8ux + Lux + XPux = f

converges strongly in L2(R"), relations:

(2.10)

(2.11)

when X goes to infinity, to the limit u defined by the

u= 0

8ü + Lü=f

in R"

in ßc,

iniï,

u= ü

in ß;

Px/2ü\aaEM(p-x/2(A„)p-x/2).

Henceforth we will assume that the operator L depends also on the parameter

t E [0, F] and we will write

L(t) = AQ(t,x)y( + 2Aj(t, *)¿

+ B(t, x)

where Aj and B are k X k matrix-valued functions on [0, T] X R" with Aj hermitian symmetric and D"xAj, D?XB bounded and continuous on [0, T] X R" for | a |< 3, | ß | < 2, 0 8xI>0

V(t,x)

E [0, T] XR".

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382

CLAUDE BARDOS AND JEFFREY RAUCH

We consider two problems involving a large parameter X:

(2.12)

(L + XP(t,x))ux=f

in[0, F]XR"

and

(2.13)

[L + XÄ0(t,x)(d/dt)]ux=f

in[0, F] X R"

with Cauchy data (2.14)

Ux(0,.)=g(.)

in{0}XR",

suppg(-)Cß

where Ä0 and P are uniformly positive matrices on ßc and vanish in ß. In either case, ux is uniquely determined and ux -> 0 on ßc. The problem is to characterize lim ux = m as the solution of a boundary value problem on [0, F] X ß. We have

Theorem 3. Assume that the matrix-valued function P has the following properties:

P — 0 on ß and there is ay > 0 such that P(t, x)>yl V(t, x) G [0, F] X ßc.

(2.15)

F G C'([0, F] X ߣ) and for \a\=

l,D,axP

is bounded on [0, F] X ßc.

Assume that fG L2([0, F] X R"), g G L2(R"). Denote by ux the solution of the Cauchy problem (2.12), (2.14) by ü the solution of the problem

\LU=f 1

'

in[0,T]xQ,

«(0, •)=«(•)

\Px/2UEM(p-x/2ArP~x/2)

inQ,

m[0,F]X3ß,

and by u the function \0 ' u(x)

ifxEQc, ifxEÜ.

Then as X - oo, ux -> u in C([0, T\\ F2(R")). To study the limit in problem (2.13) we will have to impose positivity regularity hypotheses analogous to (2.15). Precisely we assume

Positivity. Ä0(t, x) = 0if;cGß

and

and there is a y > 0 such that Ä0(t, x) s* yI if

xEÜc.

Regularity. For | a | < 2, D"XÄ0is uniformly bounded on [0, F] X ßc. However, reflecting the fact that the perturbation in (2.13) is more singular than in the other problems, we will also have to restrict the manner in which Ä0 and Av vary with time. The result we obtain is false without some such restriction, as we show by example in Remark 6 following the statement. Restricted

dependence

on

time.

For

all

(t, x)

with

x E 3ß

the

matrix

(Ä0)]/2(Ä0 1/2); maps the eigenspace M(Ä0,/2A„ÄqX/2) into itself. Let K_ be orthogonal projection onto the complement of this space. For x E 3ß, dK_/dt also maps M(ÄqX/2AvÄ0x/2) into itself. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

MAXIMAL POSITIVE BOUNDARY VALUE PROBLEMS

383

This condition is surely satisfied if Ä0 and Är do not depend on time. More generally if there are matrix-valued functions R0(x) and Rx(x) on 3ß, scalar valued functions c0, c, on [0, T] X 3ß all smooth with i0r [0,F]x3ß

= c0(/,x)R0(x),

Aj[0,T]XdQ

= cx(t,x)Rx(x),

Then on 3ß, (i0)1/2(i0"1/2), = cJ,/2(3co 1/2/3/)/ and K_ is independent of i. This example gives the spirit of the restriction; it is a restriction on the amount of twisting. Theorem 4. Assume that Ä0 satisfies the positivity, regularity and restricted dependence on time hypotheses above. Assume that f E F2([0, T] X R") and g E L2(R"),

g|a « m C([0, F]; L2(R")), w/iere u is defined by the relation

u —0

z/x G ßc,

u — ü if x E ß.

Remarks. 1. Theorem 1 has been considered in [1] but only the weak convergence is proved there. Strong convergence and the Hx/2~v estimates were proved by Kato [7] in less generality (assuming that the matrix v ■A is nonsingular on the boundary). 2. Theorems 2 and 4 provide proofs for two results predicted by Friedrichs in [4]. 3. In the case A0, Aj, B and F do not depend on t. Theorem 3 follows from Theorem 2 by a variant of the Trotter-Kato convergence theorem (see [10, p. 52] for a similar argument). In the same vein one could consider the parabolic system

^-^4^

+ ^ + ^^ "el[o,r]x3Q = 0.

m[0,F]Xß,

"e(0,-)=g(-)

as e —■0. If the coefficients are independent of t, the Trotter-Kato theorem together with Theorem 1 describes the limit. If the coefficients depend on /, a separate argument is required. This problem will not be discussed though no essentially different ideas are required. 4. The problems discussed in Theorem 4 pose some interesting technical difficulties. For example, because of the discontinuity of A0 + XÄ0 on 3ß, the term

(A0 + XÄQ)du/dt is not a well-defined distribution for u E L2([0, T] X R"). Since the large coefficient is in the terms of highest order, this perturbation is, in a sense, more singular than the others. In addition, it is interesting to observe the fundamental importance of the coefficient of 3/3? in the physical applications. It not only defines the energy but in many problems is the only coefficient which depends on x

2The existence and uniqueness of the solution of (2.13), (2.14) are discussed in §3. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

384

CLAUDE BARDOS AND JEFFREY RAUCH

and t. The latter point is emphasized in Wilcox [16] where one can find many examples. Using these as a starting point a variety of examples illustrating Theorems 1-4 are easily constructed. 5. The physical mechanism excluding waves from [0, F] X (R" \ß) is very different in Theorems 3 and 4. In Theorem 3 the principal part of the operator is unchanged so that the basic objects of geometrical optics, sound speeds and rays, are not affected by the perturbation. What one has is very rapid dissipation with a half life of the order l/X in the region R"\ß. For the more singular problems in Theorem 4 the sound speeds 0 such that

(2.18)

support ux E {(t,x):

dist(x,ß)

Waves travel very slowly

< C\t\/X).

Thus even though there may be no loss of energy in ßc the waves are effectively excluded. 6. We show by example that the condition restricting the time dependence of A0 and /!„ cannot be entirely omitted. Consider the system

where P(t) is a nonsingular positive symmetric matrix-valued function specified below and Xf signifies the characteristic function of F. Then

Ä0 = P-2,

AV= P-\XQ

_«)/»-•,

i0-V2^i-./2=(l

to be

_0J

According to Theorem 4 the expected limiting boundary condition is that the second component off'« vanishes when x = 0. We will show that for a suitable choice of P this limiting condition is not satisfied, that is

second component (P~xux(t,Q))

-«0

For this example the restriction that Äx0/2d(Ä0l/2)/dt

asÀ^

ce.

maps M(Ä0x/2AiiÄqX/1)

into itself will be violated. The substitution ux = Pvx transforms the system to

(i + ax.o.oo,)^ + ( J

_°,)^r

+ tatW-ty

We choose

p(o=«p(°,7), so

p~ip.=(°x "¿I'

/,(°)= 0-

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= o.

MAXIMAL POSITIVE BOUNDARY VALUE PROBLEMS

385

Then P = P* and F(0) is positive so we may choose 0 < F so that F is positive for

0 < t < T. For Cauchy data we take VX(0,X) = (X[-7\0](*).0)-

It is not difficult to show that the solution vx = (v\, v\) on [0, F] X R satisfies

v'x>0,

i =1,2,

vx > 1 ifx-i/(l

+ X) XX,r0](x-t/(l+X)).

Integrating from time zero yields

v{(t,0) >\t/2(l

+X).

In particular, it is not true that vx -* 0 as X -» oo. Thus ux provides the desired counterexample. Notation. We will use lower case letters e¡¡, bk, etc. for the functions obtained after localization and change of coordinates, use £>•for d/dx, in these cases and eventually use the standard summation convention instead of the sign 2. Finally we will use the same letter c for several constants, all of them independent of e and X. 3. Basic a priori estimates and weak convergence. It is known (see [1] for example) that the solution ue of the Dirichlet problem (2.7) lies in Z/2(ß) D //'(ß) and, for

8 s* 80 + 1, satisfies (3.1)

eö\\vuJ\2LHa) + 82\\ue\\2L2m (8 - 80) ( | ux(x) |2 dx.

Since P is strictly positive on ßc this implies that for 8 > 80 + 1,

(3-3)

S2\\ux\\h^)

+ X8\\ux\\2L,(Q,) 0 such that

(3.6) MI«x(/)lliJ(0«)+ll«x(/)lli^)^c(||/|||2(S)+

Hglll^

+ AllgllV,).

Sketch of proof. For u E 0 we can prove (3.6) by taking the scalar product (in L2([0, t] X ß)) of (3.2) with u and integrate by parts to obtain (A0u, u)a \'0= Re(w, /)[o,í]xa

+ (A,u, «)[0-í]x3ñ + (Zu, w)[o,\

by

in L2(R").

= g

'

where P(x) — 0 if x E R"_ and P(x) = I if x E R"+ , and where g denotes a function bounded in L2(R") uniformly in X. If K denotes the orthogonal projector on the negative subspace of the matrix p~]/2axp~x/2, then d(K^vx)/dxx belongs to the space L2(R; H](R"~X)) and, therefore, (K_vx)(0, ■) is defined (at least in //~1/2(R"-1)). Furthermore, DX(K vx) |R is uniformly (with respect to À) bounded

in L2(R_; //"'(R""1)), and K_vx is uniformly bounded in F2(R_ ; L2(R"-')). Therefore, introducing a new subfamily we deduce that we have:

(3.10)

vx-v inL2(R_ X V K_vx(0,-)^K_v(0,-)

;L2(R"-')), V " in//-'/2(R"-1).

For xx < 0 we put p(xx, x) = p(-xx, x). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

389

MAXIMALPOSITIVE BOUNDARY VALUE PROBLEMS

From (3.10) we deduce that ux^ü= p~x/2v in L2(B) and K_px/2ux |V|=0 converges to K_ px/2ü \x =0 in H~x/2(Rn~x). Finally we will show that K_px/2ü\x =0 = 0 or, equivalently, K_v(0, •) = 0. It suffices to prove that A^ux(0, •) converges to zero in H~X(R"~X). In R+ we have

(3.11)

K_p-x/2axp-x/2K_Dx(K_vx)

+ \K_vx

= hx

where hx is uniformly bounded in the space L2(R+ , H~X(R"~X)). The operator K_p~x/2axp~x/2K_ restricted to K_(Ck) is strictly negative. Therefore we deduce

from (3.11) that we have (with a different hx) (3.12)

-Dx(K_vx)

+XßK_vx

= hx inR+XR""1

where ß is a smooth and strictly positive matrix. Denote by A the Laplacian in the tangential

variables x2, x3,...,xn.

L2(R+; //'(R"-'))

For ¡x > 0, jti — A defines an isomorphism

onto L2(R+ ; //"'(R""1)).

of

Define wx by K_vx = (¡i - k)wx.

Multiply both sides of (3.12) by wx dx2 dx3.. .dxn and integrate over R"~ ' to obtain (/Ix.m,a)l2(R"-1)

(3.13)

= (XßK_vx,

,

wx)í.2(R,-.) - D\K_vx,(ii

,

„,_,

- A)

,

K_vx)LHR„-ly

It is not difficult to see that for ¡xsufficiently large one has (XßK_vx,wx)

L2(R»-i)= (A/5(it-Â)n'x,wx)L2(R„-,)>XY!lH'xll2ïi(R,.-i)

where y is a strictly positive constant. Thus (3.14)

-\DX || wx || Jf.pp-ij + Ay || wx\\ 2/1(R„-,)< (hx, wx)i.2(R«-,)

where the right-hand side is bounded in LX(R+ ) since hx is bounded in L2(R+; //-'(R"-1)) and wx is bounded in F2(R+; HX(R"~X)). Integrating (3.14) over R+ yields \Wwx(0, •)II2/i(R-i) + AyHh'xIIÍ2(R+;//1(a^I))x(0, -)\\H- '(r«-1) < c||wx(0, -)|| wi(R»-i) goes to zero when X goes to infinity, so K_px/2ux \x =0 converges to zero on H~X//2(R"~X), and the proof of weak convergence in Theorem 2 is complete. The proof of the weak convergence in case of Theorem 3 is similar; in fact, the

operator L(t) = A0(t, x)(d/dt) + lAj(t, x)(d/dxj) + B + XP defined in L2([0, F] X R") is of the same type as the operator 2Aj(d/dxj) + B + XP defined in L2(R"). The only difference is that the domain [0, F] X ß has a corner and that the boundary condition is not of constant rank there. If ¿7is a weak limit point in L2([0, T] X ß) of ux |[o,r]xo we must show that ¿7is a weak solution of the boundary value problem (2.16). Precisely we need to show that

ff

J J[o,T]xa

(u,L* 0 we find that for X large 3fx

(3.17)

X^-^

i/o

i/i

+ + aö'/W/2/Vx àëx/2axàôi/2Dxvx

,

, ., 3añl/2

+ + \äö^2^—vx Xä0 W2-^r

lies in a bounded subset of F2([0, F] X [xx > 0}). Let K (t, x) be the orthogonal projector onto the negative eigenspace of ¿¡Ôx/2axä~Q 1/2. Taking the C* scalar

product (3.17) with K_ vx we find that \jt

(3'18)

| K. vx |2 + Dx (K_ vx, 5" x/2axä-0x/2K_ vx)

J ,/,9«o"'/2 \ , / 9*+ X^^x,aö'/2-^— vxJ + 2XRe{vx,K_—vxj

lies in a bounded subset of L'([0, F] X {xx > 0}). Letting K+= I - K terms in (3.18) are written as

(3.19) \lK_vx,ä0^^K_vx\

+ \Uvx,K_äöx/2^^K+vxY

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\

the last two

391

MAXIMAL POSITIVE BOUNDARY VALUE PROBLEMS

and

(3.20)

ZK_ \ „_ / dK 2XRe(K_vx,1 dt-K_vx) K_vx, K_ '^-K+v "-"*/ + 2XRe( -.*"-"*.»"g,

The first term of each of these is no larger than cX | K_ vx \2. It is in estimating the last terms in (3.19), (3.20) that we use the restrictions on the time dependence of Ä0 and Av. As observed in (2.18) the vx vanish for xx ** c/X and the matrices ä0, 3 0} are bounded by

cXf f J0 Jx,»0

-rpi

To take advantage of this we set

Qx(t) = \jxjK^(t,x)vx(t,x)\2dx\

.

Note that g(0) = 0. Integrating (3.18) over [0, t] X {xx > 0} then yields

Aßx(0(3.21)

ff

(äni/2axänx/2K_vx,K_vx)dtdx2...dxn

° {^°}

cXf'Q2x(s)ds + cf'Q(s)ds. Ja Ja Notice that

(3.22)

(äöx/2axäZx/2K_vx,K^vx)
0]) = O(X-x/2)

and the proof is complete.

4. Tangential regularity. The main goals of this section are to show that as e — 0 or A -» oo the "tangential derivatives" of the solutions to our problems can be estimated independent of e and X.

The Dirichlet problem (2.7) is best known. If f E //^ß)

for -1 0

HX/Wll«i11(B)=H*I^1«'HhL(«-)+

2 H^*",llffL(»_)+

H«'!!!1«.)-

fc= 2

Theorem 5. There is a 8X> 8n and c > 0 so that if8>8x,X>0 (1) ///

G //^(ß)

and e > 0 then:

and ue is the solution of (2.1) which arises in Theorem I, then,

^IIVMjl2îL(a) + 52||«e||2ïL(a)0).

k= \

Assume that p(x) is a hermitian matrix-valued function with the properties:

p(x) = 0 ifx E R"_;_ p(x) |R» belongs to C2(R"+ ) and is strictly positive.

Thenfor any e > 0, 8 > 0, X > 0, the equation

(4.5)

-e[Dl(eiJDJv)

+ bkDkv] + akDkv + 8v + Xpv = f

can be reduced to an equation of the same type but with "leading" coefficient ax independent ofx. More precisely, there exists a matrix V (V E C3(R"), V~x E C3(R")) such that v = V~xv is a solution of the equation

(4.6)

-eDfâjDjv)

+ ebkDkv + äkDkv + (8q + rx + er2)v + Xpv = V*f

which has the following properties: (1) the matrices a¡ are hermitian symmetric;

(2) the operator £>-(e(-■£),-)is elliptic in the sense o/(4.4); (3) the matrix q is strictly positive; (A) the matrix p is zero on R"_ and is C2(R+ ) and strictly positive in R"+; (5) the matrices ëtJ, bk, äk, q, r¡ are smooth (belong to C2(R")) and äx is independent

of x. Proof. We introduce the matrix }¡a+(x),

(4.7)

V(x) = U(x)

0, 0.

0,

J-a_(x), 0,

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0

0

397

MAXIMALPOSITIVE BOUNDARY VALUE PROBLEMS

then we have

//, (4.8)

äx(x) = V*(x)ax(x)V(x)

=

0,

0'

0,

-/,

0

\0,

0,

0

Now we put v = Vv and we multiply both sides of (4.5) by V*. The system takes the form (4.6). The properties (l)-(5) are readily verified. D The next result gives the basic a priori estimates for the tangential derivatives near 3ß. The proof is rather technical; the main ingredients are: (1) the localization of Lemma 3, and (2) the observation that the localized operators nearly commute with tangential derivatives in the sense that the commutator involves only first order derivatives in the tangential directions. Lemma 4. Assume that the coefficients e¡¡, ak, bk, p satisfy the hypotheses of Lemma 3. In addition suppose that a0 E C2([0, F] X "35:Hom^C*)) is hermitian and strictly positive and that â0 is a C2 hermitian strictly positive matrix on [0, T] X "3o+ , and that â0 = 0 for x E "S_ . Then there are constants 83 > 80 and c > 0 so that for all X > 0,

8>83and0< e: (1) Ifv E H\R"

) n HX(R" ), supp v E %_ satisfies

(4.9)

-eDje.jD^

+ akDkv + 8v = g,

then for i = 1 or i = 2,

(4.10)

eSIIVv II2„,n(R„+ 821|v II^}

< Ilg II^.

(2) Ifv E C°°(R"+) D C°°(R"_) n C(R"), supp v E satisfies (4.11)

akDkv + Xpv + 8v = g,

then

(4.12)

S2IMIk„(R") + MUv\\%U9+) < c||g||2„L(Rn).

(3) If v E C°°([0,F] XRL) n C°°([0,F] XR"+) n C([0, F] X R"), supp v C [0, F] X (t) *£c$(0) + c/o'||g(,)||2„L(in

+

dt[s)

ds. L2(R")

Proof. We prove (1) and (4). The proofs of (2) and (3) follow the same lines and are somewhat simpler. Let v be the solution of (4.9). We apply Lemma 3 to cast equation (4.9) in the

form n

(4.18)

-eDiëijDjV + äxDxv+ 2 äkDkv + 8qv = g k= 2

where äx is independent of x and

(4.19) •• /

»gil,,, , we need estimates for || Dr> \\^j2, 2 < r < n, ' and \\xxDxv \\Li. Let D represent either the operator xxDx or Dr for r > 2. Differenti-

ating (4.18) yields (4.22)

Me(Dv)=Dg+[Me,D]v.

Taking the real part of the L2 scalar product with Dv and using (4.21) to estimate

e[v, D]v yields for 8 large (4.23)

e\\Dvv\\2L2

+ 8\\Dv\\2L2^c\\g\\2HL

+ Re(Dv,[Mc,D]v).

We will show that for each D,

(4.24) |Re(Dt3,[ME,/)]F>ü)|2e„)D1t3||L2,t5, eTDfv) \. The symmetry of T implies that

Re(/>,£>, TD2v)={Dx(Dxv,

TDxv)~

(Dxv,(DxT)Dxv).

Integrating over R"„ the integral of />,( ) vanishes since T = 0 when xx — 0. Thus |Re(D,t5,£rD2t3)|(0/f,,„(R">

3t5 + M\v\\hUxK) + X 3/

¿2(R"+)/

Let Qx be the differential operator (ä0 + Xä0)(d/dt) + 2"xakDk; then the basic energy estimate for Qx is Vw G C'([0, F] X R"_) n C'([0, F] X R"+) with äxw E C([0, F] X R") and supp w E [0, F] X %.

(4.28)

r](t)^C71(0) + cf'i1(s) + \(w(s),Qxw(s))\ds, •'o

0
rt3for 2 =£ r < n, dv/dt, and x,Z>,t3. Differentiating equation (4.26) with respect to xr for 2 < r =s n or with respect to t, one finds

ßx(F>t5) = Z)g+[Ox,F>]t5 where D = d/dt or D — Dr, 2 < r =£ n. Since 5, is independent tor satisfies | (v(s), [Qx, Dr]v(s)) | «= c$(s), so by (4.27),

of t, x, the commuta-

(4.30) HÇ?x(/)t5)(5)||22(in < c*(í) + llg(^)ll^L(R»,+ 119^(0/9^112^) where $ is defined in part (4) of the lemma. Notice that äxDv = Däxv is continuous on [0, F] X R" so we may apply the basic energy estimate (4.28) with w = Dv. Similarly, differentiating equation (4.26) with respect to xx and multiplying by xx

yields Qx(xxDxv) = xxDxg+[Qx,xxDx]v. Notice that even though Dxv need not be continuous on R", xxDxv is and we may apply the basic energy estimate (4.28) with w — xxDxv. Toward this end observe that I (xxDxv(s),

[Qx, xxDx]v(s))

|< c u in

C([0, T\. L2(ß)) for/G L2([0,T\: HlJR")) n Hx([0,T]: L2(R")) and g G //'JR") with glßc = 0 since these are dense subsets of L2([0, F] X R") and {g G L2(R"): g \QC= 0), respectively. Then by Theorem 5, {ux: X > 0} is bounded in

C([0, T\. HxtJRn)) n C'([0, T\. L2(R")). As in the proof of Theorem 2, Avux-Avu weakly in H~x/2((0, T) X 3ß), and the tangential regularity implies that {Avux}x is bounded in Hx((0, T) X T) where T is a compact neighborhood of 3ß. Thus {Avux} is bounded

in H]/2((0,

T) X 9ß), hence precompact

in F2((0, T) X 3ß) so Avux -*

Avu strongly in L2((0, F) X 3ß). For this problem, the energy identity in [0, /] X ß

for 0 < t < F yields UK - u)(t)\\20w 0. This result is sharp in the sense that if s > \ then wEwill not, in general, be bounded in /F(ß). To see this, observe that if uc were bounded in /F(ß) we would have wE— u in /F(ß) so the trace theorem yields ut |aa ^ u |9ß in /F~1/2(3ß). Since License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

MAXIMAL POSITIVE BOUNDARY VALUE PROBLEMS

403

ue |3fi = 0 it follows that u solves the boundary value problem (L + 81 )u = f in ß,

u[àa = 0. Unless Av(x) ^ 0 for all x E 3ß, there exist f E L2(ß) for which this boundary value problem will have no solutions. Thus, except when Av is negative, we cannot expect to have {ut: e 3=0} bounded in /F(ß) for any s > {. To see whether there is boundlessness in Hx/1 we consider the special case

(eD2 + Dx)ue = 1,

«£ = 0,

0(ü) and therefore is precompact in L2(ß). Since ue -* u weakly in L2(ß) it follows that ue -> u strongly in L2(ß).

If ß is not bounded, choose 2, WajDjw] || //'/2-,(R_;Z.2(Rn-,)) < ell /1| „2an(a).

The üjDjW®part of OjDjWis written as (note Djp0 — 0) ajDjW? = ajDjP2w° = ajPoDjW?.

Thus, multiplying (5.11) by p0 yields n

-e/)2vwE°+ 2 PaCijP0DjW° + 8p0ex-xxp0w?+ p0bw°

(5.13)

' ,

"

\

= Po\K + K- **r>.' - H' - 2 ajDjwl .

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MAXIMALPOSITIVE BOUNDARY VALUE PROBLEMS

407

Since p0 is just projection on the first / variables, we may let v = ((we0),,... ,(w°)¡)

to obtain v E HX(R_ XR") n /F2(R_ XR") and n

(5.14)

-e£>2t; + 2 rj(x)DjV + 8q(x)v + b(x)v = g 7-2

where g is the first / component of the right-hand side of (5.13), r, b, and q are C2(R": Hom(C')) with r and q hermitian, q positive definite and all independent of x for | * | large. We use two estimates for the solutions to (5.14), both valid uniformly for 0 < e and 8 large, (5.15)

5|l«ll//l/2-,(R.Z,2)QL.:L>)*cWg\\&VU:L*y

This is proved using the multiplier D2v. Interpolating

between (5.17) and (5.18)

yields

(5.19)

8lle||„,